-a project in the course Black Holes at Chalmers University of Technology
by Fredrik Berndtson, John Gunnarsson and Johan Johansson
Introduction
In many science fiction books and films one often find the story of people travelling backward or forward in time. How many haven't seen the film "Back to the Future", where Michael J. Fox travels back in time to where his mother and father where young and then his mother falls in love with him instead of his father and therefore he ceases to exist. That is science fiction but what is really allowed if one take into account the Law of Physics? Is it possible to travel in time and is it possible to change the past and therefore also change the present? We should here try to give a short description of what physicist know about time travels today. To be able to talk about time travels we will first have to explain what a wormhole is.
Description of a wormhole
A wormhole is a geometry of four-dimensional spacetime (for an explanation of spacetime see "spacetime" and "spacetime diagrams") in which two regions of the universe are connected by a short narrow throat. A classical large scale wormhole is a solution of the Einstein's field equations, which governs the curvature of spacetime. The most interesting thing with wormholes is that they could provide relatively easy means of travelling to distant regions of space or even of travelling backwards in time.
Figure 1. A wormhole with two mouths making a shortcut in spacetime.
The problem is that a macroscopic wormhole is not a static structure, itīs rather a shape that expands from a singularity with zero throat radius to maximum radius and then shrinks back to a singularity again. This expansion-reduction of the radius would be very quick. Even light would not have a chance to pass through the wormhole before it shrinks back to zero radius again. In fact any now known matter that would fall into the wormhole would pull it together through gravity. If constructing a mathematical model of an open wormhole that allows passage, the equations of general relativity says that matter with an enormous negative pressure is needed to uphold the wormhole gravitationally. The magnitude of the tension of the matter must be greater than the energy density of the matter itself. This would leave us with a material that will have a negative energy density relative to a light beam travelling through it. This kind of material is called exotic matter because there is no such matter now known. There are some indications that exotic matter can exist. For example between two metal plates there can be field fluctuations that has a negative energy density relative to the field fluctuations in free vacuum. Evaporating black holes also implies that exotic matter can exist.
Another problem with wormholes is that fields can destroy them. Fields are in some solutions able to increase its strength for each passage through a wormhole. If the wormhole has a focusing effect, the total field strength becomes infinite and will therefore destroy the wormhole. If the hole is defocusing, it converges towards a finite value and the wormhole can survive. The only matter that can make the wormhole defocusing is exotic matter. If exotic matter exists and has the ability to uphold a wormhole without interacting with and harming the traveller, then there is a physical possibility of a traversable wormhole and for it to even work as a time machine. A wormhole can be turned into a time machine by keeping one mouth of the wormhole fixed while moving the other. This can be made through gravitational attraction or by charging it electrically while moving it with electric fields. Travelling from the stationary mouth to the moving and back again could then send a traveller back in time.
How time travel is possible, "the Twin Paradox"
It is easy to see how one can make a time machine if one consider the "twin paradox" in special relativity (for an explanation of this paradox see "Explanation of the twin paradox"). Let an observer A be fixed in a frame and let B be another observer moving with (high) velocity u relative to A. The clock moving with B is then going with a slower rate than a A's clock because of the time dilation in special relativity. One can write an expression that relate both time as: T=gamma*T' where gamma is the Lorentz factor:
Since gamma is always > 1 B's clock is going slower than A's. To have a time machine A and B must be able to hold on to one of the wormholes mouths each. A and B can then communicate either through space or through the wormhole. A message sent through space travels with the speed of light while a message sent through the wormhole takes a shortcut in spacetime. A message sent through the wormhole will therefore arrive almost at once if the wormhole is short. In figure 2 one can see a spacetime diagram of the situation.
Figure 2. Spacetime diagram of a possible time machine.
In the figure one can see how B's clock goes slower than A's. If A sends a message to B at t = 0 through space and B replies through the wormhole the message will return to A at a time t larger than zero. But because time dilation add up while B is moving relative to A there will after a while be a time when the A observer receives the message at the same time he sends it. This is the so called "". After that time are formed and A receives his message before he sends it. This implies that time travel is possible, but of course you have to find a macroscopic, stable wormhole to be able to travel through it and you must also be able to control its mouth. One sees in the spacetime diagram that it isn't possible to go back to times before the time travel boundary was formed. This is a severe limitation of such time travels.
The Principle of Self-Consistent Solutions, "the Grandparent Paradox".
One of the problem with time travel is the so called "grandparent" paradox. Suppose you are moving backwards in time and kill your grandparent before he or she has any children. Could you do this and not, in doing so, eliminating your own existence? And even if you do that, then no one killed your grandparent in the first place and your birth is possible again! You could form a postulate that states that these events are forbidden, but that would in a sense mean that you have no free will since you are not "allowed" to kill your grandparent then. The essence of this problem is pinpointed in the billiard ball analogy. Suppose you have a billiard table with two holes. These holes are connected by a wormhole.
Figure 3. Self-consistent and not self-consistent solutions to the billiard ball problem
If you shoot a billiard ball towards and into a wormhole and it is shot out of the other mouth, but at an earlier time (due to hole's time travel effect) and then collides with the billiard ball before it reaches the first hole, it would change its direction so much that it doesn't reach the hole in the first place. Is such a solution allowed? If it is we have a real paradox. This is illustrated in figure 3A. If you change the initial conditions slightly you often receive an infinite number of solutions but not all of them are self-consistent. Self-consistency means that an event is not allowed to change the past and that the future already has affected the past! This means that we are able to neglect all the solutions when the first ball misses the hole due to the collision with the second. The solutions listed below and in figure 3A-F above and they all use Newtonian mechanics everywhere except in the wormhole that is linking the two mouths.
I will from now on call the ball that we have in our hand before we begins as the "first" ball and call the time-travelled version for the "second". If the second ball hits the first ball at just a slightly different angle and only giving it a glancing blow so that the first still will fall into the hole, but at a different angle than it had "before", then the solution will be self-consistent (figure 3 B). In the figure there are demonstrated a number of different solution classes. The figure D is one of the most interesting. There the first's vector is missing the hole altogether, but the second ball appears from one of the holes and collides with the first, and by doing so forces the first into the hole and thereby enables it to collide with "itself". This is quite similar to the fluctuations of quantum mechanics and the probability for such an event has a certain value, and are therefore not necessarily forbidden. The most common solutions are the ones in figure 3 E and F.
All these different solutions are just different approaches to the "true" grandparent paradox. But this is not really a paradox, as it first seems. It seems that the past is already set, but so is the future! But this limits the free will of a person, since he is not allowed to kill his grandparent due to the laws of causality (for an explanation of causality see "causality"). But these restrictions of free will are a common part of our daily life. Even if you want to walk on the wall, gravity prevents the efforts to do so. So it is no paradox and the rules of physics are preserved. This gives us reason to believe that time travel may be physically more possible than we perhaps may think, although there are a lot more to learn in the subject.
Conclusions
We have seen that time travels, in the way that is given above, doesn't violate the law of physics, although it seems that we cannot realise it in the nearby future. We are in fact not even sure that exotic matter exists and if it exists it isn't very probable that we can collect enough of it to feed a wormhole. We also have to find a macroscopic wormhole to feed with this exotic matter. The problems are many and if it really is possible it will take many years before we overcome them. Physicists don't agree if time travel is possible. Stephen Hawking wrote in 1993: "... the best evidence we have that time travel is not possible, and will never be, is that we have not been invaded by hordes of tourists from the future". Time travels might be allowed theoretically but real ones are still science fiction.
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